Find the term independent of x[4x^3-2\x^2]^5
To find the term independent of x in the expression [4x^3-2x^2]^5, we need to expand the expression using the binomial theorem and identify the term with x exponent equal to zero.
Using the binomial theorem, the expression [a+b]^n can be expanded as:
[a+b]^n = C(n,0)*a^n*b^0 + C(n,1)*a^(n-1)*b^1 + C(n,2)*a^(n-2)*b^2 + ... + C(n,n-1)*a^1*b^(n-1) + C(n,n)*a^0*b^n,
where C(n,k) represents the binomial coefficient "n choose k".
In our case, a = 4x^3 and b = -2x^2, and n = 5. Plugging these values into the formula, we get:
[4x^3-2x^2]^5 = C(5,0)*(4x^3)^5*(-2x^2)^0 + C(5,1)*(4x^3)^4*(-2x^2)^1 + C(5,2)*(4x^3)^3*(-2x^2)^2 + C(5,3)*(4x^3)^2*(-2x^2)^3 + C(5,4)*(4x^3)^1*(-2x^2)^4 + C(5,5)*(4x^3)^0*(-2x^2)^5.
Simplifying each term, we have:
[4x^3-2x^2]^5 = 1*(4x^3)^5*1 + 5*(4x^3)^4*(-2x^2)^1 + 10*(4x^3)^3*(-2x^2)^2 + 10*(4x^3)^2*(-2x^2)^3 + 5*(4x^3)^1*(-2x^2)^4 + 1*(4x^3)^0*(-2x^2)^5.
Now we can evaluate each term:
[4x^3-2x^2]^5 = (4x^3)^5 + 5*(4x^3)^4*(-2x^2) + 10*(4x^3)^3*(-2x^2)^2 + 10*(4x^3)^2*(-2x^2)^3 + 5*(4x^3)^1*(-2x^2)^4 + (-2x^2)^5.
Simplifying further, we get:
[4x^3-2x^2]^5 = 1024x^15 - 640x^10 + 240x^5 - 48x^10 + 32x^10 - 32x^10.
Combining like terms, we have:
[4x^3-2x^2]^5 = 1024x^15 - 656x^10 + 240x^5.
Therefore, the term independent of x is 240x^5.
To find the term independent of x in the expression [4x^3 - 2x^2]^5, we need to expand the binomial using the binomial theorem. The binomial theorem states that (a + b)^n = nC0 * a^n * b^0 + nC1 * a^(n-1) * b^1 + nC2 * a^(n-2) * b^2 + ... + nCn-1 * a * b^(n-1) + nCn * a^0 * b^n.
In our case, a = 4x^3 and b = -2x^2, and n = 5. Plugging these values into the binomial theorem, we have:
[4x^3 - 2x^2]^5 = 5C0 * (4x^3)^5 * (-2x^2)^0 + 5C1 * (4x^3)^4 * (-2x^2)^1 + 5C2 * (4x^3)^3 * (-2x^2)^2 + 5C3 * (4x^3)^2 * (-2x^2)^3 + 5C4 * (4x^3)^1 * (-2x^2)^4 + 5C5 * (4x^3)^0 * (-2x^2)^5
Simplifying this expression, we get:
[4x^3 - 2x^2]^5 = 1 * (4x^3)^5 + 5 * (4x^3)^4 * (-2x^2) + 10 * (4x^3)^3 * (-2x^2)^2 + 10 * (4x^3)^2 * (-2x^2)^3 + 5 * (4x^3)^1 * (-2x^2)^4 + 1 * (-2x^2)^5
Now, we need to find the term independent of x, which means the term that does not contain any powers of x. Looking at the expression, we can see that the term containing x^3 * (-2x^2)^2 = x^3 * 4 * x^4 = 4x^7 is the term with the highest exponent of x in the expression. All other terms will contain at least one power of x.
Thus, the term independent of x in the expression [4x^3 - 2x^2]^5 is 4x^7.